Generalised Krylov complexity

TL;DR

This paper introduces a set of generalized Krylov complexities for operator growth, demonstrating their universal features and establishing bounds relating them to Krylov entropy, with implications for understanding scrambling in quantum systems.

Contribution

It extends Krylov complexity to generalized notions, establishes inequalities relating them, and analyzes their behavior in fast and slow scramblers, broadening the scope of Krylov quantities.

Findings

Universal features of generalized Krylov complexities at initial and long times.

An inequality relating variance of K-complexity and generalized notions.

Different bounds for fast and slow scramblers based on K-complexity growth.

Abstract

In this paper, we studied a set of generalised Krylov complexity for operator growth. We demonstrate their universal features at both initial times and long times using half-analytical technique as well as numerical results. In particular, by using the logarithmic relation to the Krylov entropy, we establish an inequality (\ref{master}) between the variance of the K-complexity and the generalised notions which holds in the long time limit. Extending the result to finite (but long) times, we show that for fast scramblers, the K-complexity constrains the growth of generalised complexity more stringently than the dispersion bound. However, for slow scramblers, the growth rate of K-complexity is tighter bounded by the generalised complexity in the other way around. Our results enlarge the zoo of Krylov quantities and may shed new light on the future research in this field.